Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures
Several rare-earth compounds, low-dimensional organic conductors, and spin chains exhibit low-temperature divergences of their magnetic susceptibility and specific heat (non-Fermi-liquid behavior). Such divergences are often related to disordered ensembles of magnetic impurities in those systems....
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1197312017-06-09T03:04:30Z Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures Zvyagin, A.A. Makarova, A.V. Низкотемпеpатуpный магнетизм Several rare-earth compounds, low-dimensional organic conductors, and spin chains exhibit low-temperature divergences of their magnetic susceptibility and specific heat (non-Fermi-liquid behavior). Such divergences are often related to disordered ensembles of magnetic impurities in those systems. In this work the distribution function of the effective characteristic of a single magnetic impurity, the Kondo temperature, is derived. We calculate how distributions of Kondo temperatures depend on the effective dimensionality of the problem and on the concentration of impurities. 2004 Article Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures / A.A. Zvyagin, A.V. Makarova // Физика низких температур. — 2004. — Т. 30, № 6. — С. 639-643. — Бібліогр.: 39 назв. — англ. 0132-6414 PACS: 75.20.Hr, 75.10.Pq, 71.10.Hf http://dspace.nbuv.gov.ua/handle/123456789/119731 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Zvyagin, A.A. Makarova, A.V. Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures Физика низких температур |
| description |
Several rare-earth compounds, low-dimensional organic conductors, and spin chains exhibit
low-temperature divergences of their magnetic susceptibility and specific heat (non-Fermi-liquid
behavior). Such divergences are often related to disordered ensembles of magnetic impurities in
those systems. In this work the distribution function of the effective characteristic of a single magnetic
impurity, the Kondo temperature, is derived. We calculate how distributions of Kondo temperatures
depend on the effective dimensionality of the problem and on the concentration of impurities. |
| format |
Article |
| author |
Zvyagin, A.A. Makarova, A.V. |
| author_facet |
Zvyagin, A.A. Makarova, A.V. |
| author_sort |
Zvyagin, A.A. |
| title |
Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures |
| title_short |
Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures |
| title_full |
Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures |
| title_fullStr |
Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures |
| title_full_unstemmed |
Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures |
| title_sort |
low-temperature behavior of disordered magnetic impurities: distribution of effective kondo temperatures |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Низкотемпеpатуpный магнетизм |
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http://dspace.nbuv.gov.ua/handle/123456789/119731 |
| citation_txt |
Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures / A.A. Zvyagin, A.V. Makarova // Физика низких температур. — 2004. — Т. 30, № 6. — С. 639-643. — Бібліогр.: 39 назв. — англ. |
| series |
Физика низких температур |
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AT zvyaginaa lowtemperaturebehaviorofdisorderedmagneticimpuritiesdistributionofeffectivekondotemperatures AT makarovaav lowtemperaturebehaviorofdisorderedmagneticimpuritiesdistributionofeffectivekondotemperatures |
| first_indexed |
2025-07-08T16:30:02Z |
| last_indexed |
2025-07-08T16:30:02Z |
| _version_ |
1837096976861102080 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 6, p. 639–643
Low-temperature behavior of disordered magnetic
impurities: Distribution of effective Kondo temperatures
A.A. Zvyagin1,2 and A.V. Makarova3,1
1B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy
of Science of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
2Max-Planck-Institut für Physik komplexer Systeme, Dresden, Germany
E-mail: zvyagin@mpipks-dresden.mpg.de
3Kharkov State Economic University, Kharkov, Ukraine
Received January 12, 2004
Several rare-earth compounds, low-dimensional organic conductors, and spin chains exhibit
low-temperature divergences of their magnetic susceptibility and specific heat (non-Fermi-liquid
behavior). Such divergences are often related to disordered ensembles of magnetic impurities in
those systems. In this work the distribution function of the effective characteristic of a single mag-
netic impurity, the Kondo temperature, is derived. We calculate how distributions of Kondo tem-
peratures depend on the effective dimensionality of the problem and on the concentration of impu-
rities.
PACS: 75.20.Hr, 75.10.Pq, 71.10.Hf
The Kondo effect [1] is the well-known example in
which modern theoretical methods like renormali-
zation group theory, the Bethe ansatz, bosonization,
conformal field theory, etc. have demonstrated their
strength [2,3]. It describes the effects of the exchange
interaction between the spin of a magnetic impurity
and spins of itinerant electrons. The crossover from
the strong coupling to the weak coupling regime for a
Kondo impurity manifests nonperturbative effects pre-
sent in condensed matter theory.
During the last decade the interest in non-Fer-
mi-liquid behavior of magnetic systems and metallic
alloys has grown considerably. A large class of con-
ducting nonmagnetic materials does not behave as
usual Fermi liquids (i.e., with the finite magnetic sus-
ceptibility, finite value of the Sommerfeld low-tem-
perature coefficient of the specific heat, and quadratic
temperature behavior of the resistivity) at low tem-
peratures. There are several possible reasons for such
a behavior. One of the most known examples of the
non-Fermi-liquid properties is the Kondo effect for
multichannel (n is the number of channels) electron
systems: For an impurity spin less than n/2 a non-Fer-
mi-liquid critical behavior results [4]. Another reason,
which can cause the non-Fermi-liquid behavior, is
the presence of the quantum critical point [5] (i.e.,
a phase transition governed not by the temperature
but by some other parameter, like pressure, chemical
substitution, etc.). In that case fluctuations of the or-
der parameter interact with itinerant electrons and
can cause low-temperature divergences of thermody-
namic characteristics.
However, for most of nonstoichiometric conductors
in which the non-Fermi-liquid behavior has been ob-
served, see, e.g., the recent reviews [5–8], the mag-
netic susceptibility and low-temperature specific heat
usually manifest logarithmic or weak power-law be-
havior with temperature, while their resistivity de-
creases linearly (or with some power-law exponent
less than 2) with temperature, showing a large resid-
ual resistivity [9–19]. This is different from the pre-
dictions of the theory of the overscreened Kondo effect
[3,4]. For example, Ref. 14 reported the results of
measurements of the magnetic susceptibility, NMR
Knight shift, and low-temperature specific heat. To
explain the observed features it was necessary to as-
sume some disorder, with a distribution of Kondo tem-
peratures of magnetic impurities. The inhomogeneous
distribution of localized magnetic moments was later
confirmed [18] by muon spin rotation experiments.
© A.A. Zvyagin and A.V. Makarova, 2004
The above-mentioned properties and the alloy nature
of the studied compounds suggest that the disorder (a
random distribution of localized electrons or a random
coupling to the conducting electron host) can play the
main role in the low-temperature non-Fermi-liquid
character of such systems. The idea of (unscreened)
magnetic moments existing in disordered metallic sys-
tems has been formulated in [20–24]. It was proposed
that near the metal–insulator transitions (or for the
sufficiently alloyed systems far from the quantum crit-
ical point) disordered correlated conductors contain
localized magnetic moments. The change in the inter-
actions between the impurity sites and the host spins
can be considered as a modification of the characteris-
tic energy scale, the Kondo temperature TK . At that
scale the behavior of the magnetic impurity manifests
the crossover from the strong coupling regime (for
T H TK, �� , H being the magnetic field) to the weak
coupling regime T H TK �� , . The impurity spin be-
haves asymptotically free in the weak coupling case,
and it is screened by the spins of itinerant electrons in
the strong coupling case. The random distribution of
magnetic characteristics of impurities (i.e., their Kon-
do temperatures) may be connected either with the
randomness of exchange couplings of itinerant elec-
trons with the local moments [21] or with the random-
ness of the densities of conduction electron states [20].
Both types of randomness renormalize the single uni-
versal parameter, the Kondo temperature, which char-
acterizes the state of a magnetic impurity. A thorough
comparison of experimental results for non-Fermi-liq-
uid behavior of disordered heavy fermion Kondo al-
loys has been performed with very good agreement
with theoretical predictions of the model for distrib-
uted Kondo temperatures [17]. Later it was pointed
out that the problem of the behavior of magnetic im-
purities with random distributions of their Kondo
temperatures in metals can be solved exactly,
with the help of the Bethe ansatz [24–26]. The role of
the long-range (Ruderman—Kittel—Kasuya—Yosi-
da) coupling between local moments was taken into
account [25–28] (Griffiths phase [29] theory), exhib-
iting properties qualitatively similar to those of mod-
els with noninteracting local moments. Also, the pres-
ence of a spin–orbit interaction in some disordered
heavy fermion alloys demands the study of a magnetic
anisotropy, which can play an essential role in physics
of disordered spin interactions [26–28,30]. Finally, it
was exactly proved that for correlated electrons sys-
tems with magnetic impurities with random distribu-
tions of their couplings to the host it is also possible to
introduce a distribution of effective Kondo tempera-
tures [31] which governs the low-temperature non-
Fermi-liquid behavior of the system.
It was pointed out [25,26,28,31] that distributions
of effective Kondo temperatures for each magnetic im-
purity can cause divergences of the magnetic suscepti-
bility and the Sommerfeld coefficient of the specific
heat for quasi-one-dimensional organic conductors and
quantum spin chains, where such behavior was ob-
served [32–35]. To explain power-law divergences of
the magnetic susceptibilities and Sommerfeld coeffi-
cients of rare-earth and actinide compounds, as well as
quasi-one-dimensional organic conductors and quan-
tum spin chains, it was necessary to use a distribution
of Kondo temperatures (the strong-disorder distribu-
tion, for which the tails are rather large) which starts
with the term P T G TK K( ) ( )� � �� � 1 (� � 1) valid till
some energy scale G for the lowest values of TK
[25–27,30,31]. The goal of this work is to obtain the
distribution of Kondo temperatures for a system with
magnetic impurities and to show how such a distribu-
tion will depend on the effective spatial dimensio-
nality of the system (it turns out that the exponent �
is different for three-dimensional non-Fermi-liquid
heavy fermion systems and for quasi-one-dimensional
organic conductors and quantum spin systems [16,32,
33,35]).
Let us consider the model of electrons (itinerant
and localized, where localization centers, i.e., 3d, 4f,
or 5f orbitals, are distributed randomly on a hyper-
cubic lattice with a random nearest-neighbor hopping)
with the Hamiltonian [20]
H H� � �
�
��t j j c c
j j
j j( , )
, ,
,
†
,
int , (1)
where cj,
†
(cj, ) creates (annihilates) an electron
with spin at site j, and t j j( , )� are hopping ele-
ments. The interaction part of the Hamiltonian,
H int � � �( ) , ,U/ n nj j j2 , is determined by the Cou-
lomb interaction of localized electrons (the sum is
over random positions of localized electrons, and
n c cj j j, ,
†
, � ). Generally speaking, one can add to
the Hamiltonian (1) disordered local potentials
� j j jn,
. The hopping integrals can be approxi-
mated [36] as overlap integrals
t j j E
r
a
r
a
j j j j
( , ) exp ...
, ,
� �
��
�
�
�
�
� � �
�
�
�
�
�
�
� �
0 1 , (2)
where a is the Bohr radius and E U0 ~ is the effective
binding energy of the dopant (localized electron). We
suppose that hopping integrals are random because of
the disorder of the distribution of localized electrons.
Let us consider the situation for which localized
electrons are in the magnetic state, i.e., their valence
is close to 1, which is satisfied if t j j EF
2( , ) ( )� ���
�� �
j , U j�
(with negative
j , where �( )EF is
640 Fizika Nizkikh Temperatur, 2004, v. 30, No. 6
A.A. Zvyagin and A.V. Makarova
the density of states at the Fermi level) [2]. It is natu-
ral to assume that the density of the localized magnetic
moments nl depends on the density of dopants n as
n n n/nl � �exp( )max , (3)
where nmax is related to the critical distance bet-
ween localization centers Rc via n Vcmax /� 1 . Here
Vc depends on the effective dimensionality of the
distribution of localized magnetic moments. It is
equal to V R /c c� 4 32� for the three-dimensional case,
V Rc c� � 2 for the effectively two-dimensional situa-
tion, and for effectively one-dimensional distribution
of localized magnetic moments one has V Rc c� . Such
a distribution is well-known in the theory of disor-
dered systems (it follows from simple combinatorics),
e.g., in the theory of dislocations with defects it is
known as the Koehler formula [36], see also [37,38].
For large n, nl is a decreasing function of n, as it must
be. Using such an assumption we can find the proba-
bility P r( ) to find the nearest-neighbor localized mo-
ment of a given site at a distance r, see below. (On
the other hand, P r( ) can be derived independently
[39], and we can obtain Eq. (3).) Obviously, the
density of the localized magnetic moments can be
written as
n P r drl
Rc
�
�
� ( ) . (4)
Using the definition of Rc we obtain
P r n
r nr / d
r nr d
nr
( )
exp( ),
exp( ),
exp(
�
� �
� �
�
� �
� �
2 3
2
4 3 3
2 2
), d �
�
�
��
�
�
� 1
(5)
where d is the effective dimensionality of the distribu-
tion of localized electrons, cf. [36,39]. One can check
that nl has a maximum at nmax, which, actually, jus-
tifies Eq. (3).
The next step is to obtain the distribution of hop-
ping integrals t j j( , )� between localized moments (in
what follows we shall denote them simply as t).
Equation (2) implies r t a E /t( ) ( )� ln 2 0 . Then, using
~( ) ( )( )P t P r dr/dt� and Eq. (5) we get
~( )
ln ( ) exp [ ( ) ln ( )],
P t
an
t
a E /t na / E /t d
� �
� �� �2 2
0
3 3
04 3 3
2 20
2 2
0
0
� �aln E /t na E /t d
na E /t
( ) exp [ ( )],
exp [ ( )]
� �
�
ln
ln , d �
�
�
��
�
�
� 1
(6)
for the three-, two-, and one-dimensional cases, respectively.
The Kondo temperature of the localized magnetic moment can be written as [2]
T
U E
t
U
U E t
DK
F
F
� �
��
�
�
�
�
�
� �
2
2 2
�
�
�
( )
exp
| | ( )
( )
exp
| | ( )
( )
�
��
�
�
�
�
�
U
U E tF2 2
, (7)
where we have assumed homogeneous distributions of the local potentials
j � and introduced the low-energy
cutoff D, as usual for the Kondo problem [2]. Notice that for
� 0 one can define the Kondo temperature as
T D U/ E tK F� �exp ( ( ) )2 2� . Then Eq. (7) implies
dT
dt
t
T T /D
K
K K
� �
2 ln ( )
. (8)
Defining x D/TK� ln ( ) and A U E E UF� �8 0
2�
( ) /| | ( ), we obtain the distributions of Kondo temperatures
P T
Z n
n T x
/ Ax n/ n Ax
K
d
K
( )
( )ln ( ) exp [ ( ) (
max
max
�
�
2
3 16 82 3ln )],
ln( ) exp [ ( ) ( )],
exp [ (
max
ma
d
Ax n/ n Ax d
n/ n
�
� �
�
3
4 2
2
2ln
x) ( )],ln Ax d �
�
�
��
�
�
� 1
(9)
for the three-, two-, and one-dimensional cases, re-
spectively, where Zd are normalization constants.
Observe that the divergence of P TK( ) as TK � 0 (due
to the factor TK
�1) is weakened by logarithmic factors
like ln ( )T /DK . This is why P TK( ) can be normalized
to unity over some interval 0 � �T TK max, where Tmax
is given by T D U / U E EFmax exp ( | | ( ) ( ) )� � �
�8 0
2 .
Tmax can be related to the parameter G, see above.
Obviously the distributions of Kondo temperatures
depend on the parameters of impurity
, the constant
Low-temperature behavior of disordered magnetic impurities: Distribution of effective Kondo temperatures
Fizika Nizkikh Temperatur, 2004, v. 30, No. 6 641
of Coulomb repulsionU, the density of states of itin-
erant electrons at the Fermi level, the Bohr radius,
and the density of localized orbitals. However, Eqs.
(9) imply that such a set of parameters is realized in
two main governing parameters: n/nmax and A.
It turns out that for large enough intervals of val-
ues of TK the distribution P TK( ) is proportional to
TK
��1, i.e., it is of the form used in [25–27,30,31]
without derivation, based on experimental results.
The exponent � � 1 is determined by A and n/nmax.
The plots of the logarithms of P TK( ) as functions of x
are presented in Fig. 1,a for one-, two-, and three-di-
mensional situations (we did not take into account the
terms in Eqs. (9) which do not depend on x here;
those terms yield only constant shifts). Here we used
the parameters D,
,U, E0, appropriate for real disor-
dered rare-earth and actinide systems, which produces
A � 1. Also, according to experiments, one can expect
n/nmax .� 0 01. Notice that the maximal values of
intervals of TK , Tmax, depend on A via Tmax �
� �D /Aexp ( )1 , and, hence, x /Amin � 1 . This is why
we present here the results for large enough values of
x (i.e., for small TK), that the distributions in the
considered intervals are reminiscent of the power laws
� �TK
1 �. One can see that the exponents of the distri-
butions depend on the effective dimensionality of the
distribution of the localized electrons.
To illustrate how the effective exponents depend
on A and on the concentration of localized electrons n,
we present results for the logarithms of the distribu-
tions of Kondo temperatures for A � 5, n/nmax .� 0 01
in Fig. 1,b and for A � 1, n/nmax .� 0 5 in Fig. 1,c. The
results show that the effective exponents � are large
( .� � 0 8 for the realistic case A ~ 1, n/nmax .~ 0 01 for
three space dimensions). It is too large compared to
the experiments on three-dimensional heavy fermion
systems [14,16–18], for which values of � � 0.6–0.85
were observed. Also, for some values of the parameters
a value � � 1 was obtained, which implies the absence
of low-temperature divergences in such cases. The rea-
son for such an overestimation can lie in the disorder
in the distribution of local potentials of localizated
electrons,
j , which was not taken into account in our
description. However, smaller values of � (0.26–0.42)
have been observed for low-dimensional organic con-
ductors and spin chains [32,33]. It turns out that large
values of � ~ .0 2 were observed for disordered ensem-
bles of impurities close to a metal-insulator transition.
Such a distribution of Kondo temperatures pro-
duces non-Fermi-liquid behavior of many characteris-
tics of the system [25–27,31]. For example, the
ground state magnetization of the system behaves as
M H G� ( / )� (H is an external magnetic field), i.e.,
essentially in a nonlinear way. The static magnetic
susceptibility at low temperatures is divergent as
T /G� ��1 . Also, the nonlinear behavior of the electron
specific heat c T/G~ ( )� implies divergent behavior of
the Sommerfeld coefficient. The low-temperature be-
havior of the correlation length is proportional to
( )G/T �, which implies non-Fermi-liquid behavior of
the low-temperature resistivity. Finally, the imagi-
nary part of the dynamic magnetic susceptibility is
also divergent as G T g T� � �� �2 1 ( / ), where g x( ) �
� �
�
��x dy/y x y
1
1 2 2� ( ) is a universal scaling func-
tion [25].
Summarizing, in this work we have calculated
the distribution of effective Kondo temperatures for a
642 Fizika Nizkikh Temperatur, 2004, v. 30, No. 6
A.A. Zvyagin and A.V. Makarova
2
4
6
8
3 4 5 6 7 8 9 10x
A = 1, n/n = 0.01max
2
3
4
5
6
7
8
9
x43 5 6 7 8 9 10
A = 5, n/n = 0.01max
2
4
6
8
3 4 5 6 7 8 9 10x
A = 1, n/n = 0.5max
a b c
P
(T
)
K
Fig. 1. Logarithms of the distributions of the Kondo temperatures for the three-dimensional (solid line), two-dimensional
(dotted line), and one-dimensional (dashed line) cases.
system of correlated electrons in which itinerant elec-
trons are hybridized with randomly distributed local-
ized electrons (3d, 4f, or 5f orbitals). Those distribu-
tions depend on many parameters: concentration of
localized orbitals, their Bohr radii, Coulomb interac-
tion of localized electrons on orbitals, effective ener-
gies of localized electrons, bandwidths of itinerant
electrons, etc. However, we have shown that only two
parameters effectively govern the behavior of the dis-
tribution functions of Kondo temperatures. One of
those parameters is the ratio of the density of localized
electrons to the maximal density (in a hypercubic lat-
tice). The other parameter is related to the maximal
Kondo temperatures for which such an approach can
be applied. We have shown that distributions of
Kondo temperatures depend on the effective dimen-
sionality of the disordered localized electrons. Due to
such a disordered behavior of magnetic impurities
many thermodynamic and kinetic characteristics of
the considered systems manifest non-Fermi-liquid
features in their low-energy characteristics.
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